{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T14:57:07Z","timestamp":1776265027315,"version":"3.50.1"},"reference-count":12,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2016,2,29]],"date-time":"2016-02-29T00:00:00Z","timestamp":1456704000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented by Fehribach, which can be defined independent of any matrix relations. In particular, we present simple cases in which vector graphs are guaranteed to be Kirchhoff or non-Kirchhoff. Next, we review Maxwell\u2019s method of drawing reciprocal figures as he presented in 1864, using modern mathematical language. We then demonstrate cases in which R2-vector graphs defined from Maxwell reciprocals are \u201cdual\u201d Kirchhoff graphs. Given an example in which Maxwell\u2019s theories are not sufficient to define vector graphs, we begin to explore other methods of developing dual Kirchhoff graphs.<\/jats:p>","DOI":"10.3390\/sym8030009","type":"journal-article","created":{"date-parts":[[2016,2,29]],"date-time":"2016-02-29T10:55:59Z","timestamp":1456743359000},"page":"9","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures"],"prefix":"10.3390","volume":"8","author":[{"given":"Tyler","family":"Reese","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Randy","family":"Paffenroth","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph","family":"Fehribach","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,2,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1137\/080720115","article-title":"Vector-space methods and Kirchhoff Graphs for reaction networks","volume":"70","author":"Fehribach","year":"2009","journal-title":"SIAM J. Appl. Math."},{"key":"ref_2","first-page":"125","article-title":"Matrices and their Kirchhoff Graphs","volume":"9","author":"Fehribach","year":"2015","journal-title":"Math. Contemp."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5671","DOI":"10.1021\/jp0374004","article-title":"Reaction route graphs I. Theory and algorithm","volume":"108","author":"Fishtik","year":"2004","journal-title":"J. Phys. Chem. B"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"5683","DOI":"10.1021\/jp037401w","article-title":"Reaction route graphs II. Examples of enzyme and surface catalyzed single overall reactions","volume":"108","author":"Fishtik","year":"2004","journal-title":"J. Phys. Chem. B"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.jelechem.2004.10.007","article-title":"A reaction route graph analysis of the electrochemical hydrogen oxidation and evolution reactions","volume":"576","author":"Fishtik","year":"2005","journal-title":"J. Electroanal. Chem."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1016\/j.physa.2006.05.031","article-title":"A reaction route network for hydrogen combustion","volume":"373","author":"Fishtik","year":"2007","journal-title":"Physica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/978-0-387-46108-3_3","article-title":"The use of graphs in the study of electrochemical reaction networks","volume":"Volume 41","author":"Vayenas","year":"2007","journal-title":"Modern Aspects of Electrochemistry"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"8697","DOI":"10.1073\/pnas.0602767103","article-title":"Understanding bistability in complex enzyme-driven reaction routes","volume":"103","author":"Craciun","year":"2006","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1017\/S0080456800026351","article-title":"On reciprocal figures, frames, and diagrams of forces","volume":"26","author":"Maxwell","year":"1870","journal-title":"Trans. R. Soc. Edinb."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s00158-013-0910-0","article-title":"Maxwell\u2019s reciprocal diagrams and discrete michell frames","volume":"48","author":"Baker","year":"2013","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Bollob\u00e1s, B. (1998). Modern Graph Theory. Springer Graduate Texts in Mathematics, Springer-Verlag.","DOI":"10.1007\/978-1-4612-0619-4"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Friedman, J. (2015). Sheaves on Graphs, Their Homological Invariants, and a Proof of the Hanna Neumann Conjecture: With an Appendix by Warren Dicks, Memoirs of the AMS.","DOI":"10.1090\/memo\/1100"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/3\/9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:19:52Z","timestamp":1760210392000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/8\/3\/9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,2,29]]},"references-count":12,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,3]]}},"alternative-id":["sym8030009"],"URL":"https:\/\/doi.org\/10.3390\/sym8030009","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,2,29]]}}}